Preconditioning for PDE-constrained optimization with total variation regularization

The efficient solution of PDE-constrained optimization problems is of great significance in many scientific and engineering applications. The key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning. In this paper, we consider preconditioning the linear systems arising from PDE-constrained optimization problems with total variation (TV) regularization. Based on the block structure of the coefficient matrix or its row and column transformed forms, we propose three kinds of new preconditioners by approximating the Schur complements in different ways. We analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size. We also show the relations of eigenvalue bounds corresponding to different preconditioners. Numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods.